For a great list of point values associated with actions and other useful info, check out the Ingress Field Guide.

How you go about getting AP will depend on the situation for your side in your area, but here's a few general guidelines for getting past the early levels:

Hack a lot of portals to build up an inventory (AKA farm). Friendly portals will get you overall better stuff since they give fewer "Hack acquired no items" messages, while enemy portals will give you 100 AP each time (nice for early leveling) but will drain more of your XM.

Link portals and create control fields as this yields a lot of points for early level players, heck this yields the most points per action for all level players. However, Keys drop less frequently than they used to, so even though higher level players can destroy things more easily, they often won't have the keys to link and create control fields, so if you get some keys, look at how you can use them with friendly portals to link.

Deploy shields to friendly portals. Shields are again something that drop relatively less often, so higher level players will often only place one or two on a portal, leaving space for you to come in and reinforce them

Use the map at http://ingress.com/intel to look for unclaimed portals. New portals come online occasionally that others may not know about, or there may not be active players in your area yet. To find these you have to be zoomed in until it says "showing all portals", and unclaimed portals will be grey.

Attack portals, but don't attack anything too high level unless you've got a lot of XMP bursters. In general it might be best not to attack until you've built a sizeable inventory of weapons, since it usually takes more of them than you'd think to fully take over a portal, which will yield a lot more points than destroying a resonator or two. And when you do attack, make sure to stand directly on each resonator, as low level weapons have a very limited range.

If you're in a city mostly controlled by the enemy, it can be difficult to get started. You're really going to want to reach out for friendly player help at that point. A couple people working together can get a lot done.

Even if you're not in a city full of enemy, you can level a lot faster with help from higher level players. I'll try to help lower level team mates level by destroying portals for them, and then letting them place all the resonators. An organized team is really the best way in the long run to do well in an active city. Ask in faction chat for help leveling or if there's any kind of organization. Many cities have Google+ pages or other group communication tools to discuss tactics, share hints, plan raids, share rides to farms and other things to help each other out.